Department of Physics

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Adiabatic modulation of driving protocols in periodically driven quantum systems
(ARXIV, 2024-04) Sarkar, Tapomoy Guha; Bandyopadhyay, Jayendra N.
We consider a periodically driven system where the high-frequency driving protocol consists of a sequence of potentials switched on and off at different instants within a period. We explore the possibility of introducing an adiabatic modulation of the driving protocol by considering a slow evolution of the instants when the sequence of potentials is switched on/off. We examine how this influences the long-term dynamics of periodically driven quantum systems. By assuming that the slow and fast timescales in the problem can be decoupled, we derive the stroboscopic (effective) Hamiltonian for a four-step driving sequence up to the first order in perturbation theory. We then apply this approach to a rigid rotor, where the adiabatic modulation of the driving protocol is chosen to produce an evolving emergent magnetic field that interacts with the rotor's spin. We study the emergence of diabolical points and diabolical loci in the parameter space of the effective Hamiltonian. Further, we study the topological properties of the maps of the adiabatic paths in the parameter space to the eigenspace of the effective Hamiltonian. In effect, we obtain a technique to tune the topological properties of the eigenstates by selecting various adiabatic evolution of the driving protocol characterized by different paths in the parameter space. This technique can be applied to any periodic driving protocol to achieve desirable topological effects.
Algebraic approach in the study of time-dependent nonlinear integrable systems: Case of the singular oscillator
(APS, 2001-03) Bandyopadhyay, Jayendra N.
The classical and the quantal problem of a particle interacting in one dimension with an external time-dependent quadratic potential and a constant inverse square potential is studied from the Lie-algebraic point of view. The integrability of this system is established by evaluating the exact invariant closely related to the Lewis and Riesenfeld invariant for the time-dependent harmonic oscillator. We study extensively the special and interesting case of a kicked-quadratic potential from which we derive a new integrable, nonlinear, area preserving, two-dimensional map that may, for instance, be used in numerical algorithms that integrate the Calogero-Sutherland-Moser Hamiltonian. The dynamics, both classical and quantal, is studied via the time-evolution operator that we evaluate using a recent method of integrating the quantum Liouville-Bloch equations [A. R. P. Rau, Phys. Rev. Lett. 81, 4785 (1990)]. The results show the exact one-to-one correspondence between the classical and the quantal dynamics. Our analysis also sheds light on the connection between properties of the su(1,1) algebra and that of simple dynamical systems.
Bandyopadhyay, Paterek, and Kaszlikowski Reply:
(APS, 2013-04) Bandyopadhyay, Jayendra N.
In this Reply to the preceding Comment by Gauger and Benjamin, we recognize that the numerical error was present in our software, and we explain which of our conclusions are affected by it.
Cataloging topological phases of N stacked Su-Schrieffer-Heeger chains by a systematic breaking of symmetries
(APS, 2023-09) Bandyopadhyay, Jayendra N.
Two-dimensional (2D) model of a weak topological insulator with an N stacked Su-Schrieffer-Heeger chain is studied. This study starts with a basic model with all the fundamental symmetries (chiral, time reversal, and particle hole) preserved. Different topological phases are introduced in this model by systematically breaking the system's symmetries. The symmetries are broken by introducing different bonds (hopping terms) in the system. First, the chiral symmetry is broken by introducing hopping within each sublattice or intrasublattice hopping, where the hopping strengths of the sublattices are equal in magnitudes but opposite in sign. Then, following Haldane, the time-reversal (TR) symmetry is broken by replacing the real intrasublattice hopping strengths with imaginary numbers without changing the magnitudes. We find that breaking chiral and TR symmetries are essential for the weak topological insulator to be a Chern insulator. These models exhibit nontrivial topology with the Chern number C=±1. The preservation of the particle-hole (PH) symmetry in the system facilitates an analytical calculation of C, which agrees with the numerically observed topological phase transition in the system. An interesting class of topologically nontrivial systems with C=0 is also observed, where the nontriviality is identified by a quantized and fractional 2D Zak phase. Finally, the PH symmetry is broken in the system by introducing unequal amplitudes of intrasublattice hopping strengths, while the equal intrasublattice hopping strengths ensure the preservation of the inversion symmetry. We investigate the interplay of the PH and the inversion symmetries in the topological phase transition. A discussion on the possible experimental realizations of this model is also presented.
Controlling resonant enhancement in higher-order harmonic generation
(ARXIV, 2021) Holkundkar, Amol R.; Bandyopadhyay, Jayendra N.
We present a method to tune the resonantly enhanced harmonic emission from engineered potentials, which would be experimentally feasible in the purview of the recent advances in atomic and condensed matter physics. The recombination of the electron from the potential dependent excited state to the ground state causes the emission of photons with a specific energy. The energy of the emitted photons can be controlled by appropriately tweaking the potential parameters. The resonant enhancement in high-harmonic generation enables the emission of very intense extreme ultra-violet or soft x-ray radiations. The scaling law of the resonant harmonic emission with the model parameter of the potential is also obtained by numerically solving the time-dependent Schrödinger equation in two dimensions.
Effective time-independent analysis for quantum kicked systems
(APS, 2015-03) Bandyopadhyay, Jayendra N.; Sarkar, Tapomoy Guha
We present a mapping of potentially chaotic time-dependent quantum kicked systems to an equivalent approximate effective time-independent scenario, whereby the system is rendered integrable. The time evolution is factorized into an initial kick, followed by an evolution dictated by a time-independent Hamiltonian and a final kick. This method is applied to the kicked top model. The effective time-independent Hamiltonian thus obtained does not suffer from spurious divergences encountered if the traditional Baker-Cambell-Hausdorff treatment is used. The quasienergy spectrum of the Floquet operator is found to be in excellent agreement with the energy levels of the effective Hamiltonian for a wide range of system parameters. The density of states for the effective system exhibits sharp peaklike features, pointing towards quantum criticality. The dynamics in the classical limit of the integrable effective Hamiltonian shows remarkable agreement with the nonintegrable map corresponding to the actual time-dependent system in the nonchaotic regime. This suggests that the effective Hamiltonian serves as a substitute for the actual system in the nonchaotic regime at both the quantum and classical level.
Engineering harmonic emission through spatial modulation in a Kitaev chain
(APS, 2025-09) Bandyopadhyay, Jayendra N.; Holkundkar, Amol R.
We investigate high-harmonic generation (HHG) in a dimerized Kitaev chain. The dimerization in the model is introduced through a site-dependent modulating potential, determined by a parameter 𝜆∈[−1:1] . This parameter also determines the strength of the hopping amplitudes and tunes the system's topology. Depending upon the parameter 𝜆 , the HHG emission spectrum can be classified into three segments. The first segment exhibits two plateau structures, with the dominant one resulting from transitions to the chiral partner state, consistent with quasiparticle behavior in the topological superconducting phase. The second segment displays multiple plateaus, where intermediate states enable various transition pathways to higher conduction bands. Finally, the third segment presents broader plateaus, indicative of active interband transitions. In the 𝜆≤0 regime, we observe the midgap states (MGSs) hybridize with the bulk, suppressing the earlier observed harmonic enhancements. This highlights the key role of the intermediate states, particularly when MGSs are isolated. These results demonstrate that harmonic emission profiles can be selectively controlled through the modulating parameter 𝜆 , offering new prospects for tailoring HHG in topological systems.
Entanglement and level crossings in frustrated ferromagnetic rings
(APS, 2009-04) Bandyopadhyay, Jayendra N.
We study the entanglement content of a class of mesoscopic tunable magnetic systems. The systems are closed finite spin-1/2 chains with ferromagnetic nearest-neighbor interactions frustrated by antiferromagnetic next-nearest-neighbor interactions. The finite chains display a series of level crossings reflecting the incommensurate physics of the corresponding infinite-size chain. We present dramatic entanglement signatures characterizing these unusual level crossings. We focus on multispin and global measures of entanglement rather than only one-spin or two-spin entanglements. We compare and contrast the information obtained from these measures to that obtained from traditional condensed-matter quantities such as correlation functions.
Entanglement measures in quantum and classical chaos
(ARXIV, 2005-09) Bandyopadhyay, Jayendra N.
Entanglement is a Hilbert-space based measure of nonseparability of states that leads to unique quantum possibilities such as teleportation. It has been at the center of intense activity in the area of quantum information theory and computation. In this paper we discuss the implications of quantum chaos on entanglement, showing how chaos can lead to large entanglement that is universal and describable using random matrix theory. We also indicate how this measure can be used in the Hilbert space formulation of classical mechanics. This leads us to consider purely Hilbert-space based measures of classical chaos, rather than the usual phase-space based ones such as the Lyapunov exponents, and can possibly lead to understanding of partial differential equations and nonintegrable classical field theories.
Entanglement production in coupled chaotic systems: Case of the kicked tops
(APS, 2004-01) Bandyopadhyay, Jayendra N.
Entanglement production in coupled chaotic systems is studied with the help of kicked tops. Deriving the correct classical map, we have used the reduced Husimi function, the Husimi function of the reduced density matrix, to visualize the possible behaviors of a wave packet. We have studied a phase-space based measure of the complexity of a state and used random matrix theory (RMT) to model the strongly chaotic cases. Extensive numerical studies have been done for the entanglement production in coupled kicked tops corresponding to different underlying classical dynamics and different coupling strengths. An approximate formula, based on RMT, is derived for the entanglement production in coupled strongly chaotic systems. This formula, applicable for arbitrary coupling strengths and also valid for long time, complements and extends significantly recent perturbation theories for strongly chaotic weakly coupled systems.
Entanglement production in quantized chaotic systems
(Springer, 2005-04) Bandyopadhyay, Jayendra N.
Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed-state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed-state entanglement production in chaotic systems.
Entanglement signatures of gapless topological phases in a -wave superconductor
(2025-09) Bandyopadhyay, Jayendra N.
We explore the gapless topological phases of a -wave superconductor, probing its rich topologically ordered phases and underlying quantum phenomena. The topological order of the system is characterized by studying its entanglement properties. This study confirms the bulk-boundary correspondence in the entanglement spectrum, even without a full bulk gap. For contractible bipartitions, the entanglement entropy varies non-monotonically with the chemical potential, displaying pronounced peaks at points where the bulk gap closes and reopens, signaling topological quantum phase transitions. This behavior remains robust in the thermodynamic limit. The entanglement entropy grows with system size for non-contractible bipartitions, indicating long-range entanglement in the gapless phase. These findings reveal the subtle interplay between symmetry, entanglement, and topology in gapless systems, and emphasize the role of entanglement-based diagnostics in identifying unconventional topological phases beyond the gapped paradigm.
Entangling power of quantum chaotic evolutions via operator entanglement
(ARXIV, 2005-04) Bandyopadhyay, Jayendra N.
We study operator entanglement of the quantum chaotic evolutions. This study shows that properties of the operator entanglement production are qualitatively similar to the properties reported in literature about the pure state entanglement production. This similarity establishes that the operator entanglement quantifies {\it intrinsic} entangling power of an operator. The term `intrinsic' suggests that this measure is independent of any specific choice of initial states.
European Physical Society logo. Italian Physical Society logo. EDP Sciences logo. The Institute of Physics logo. Quantum chaotic system as a model of decohering environment
(IOP, 2009-03) Bandyopadhyay, Jayendra N.
As a model of decohering environment, we show that a quantum chaotic system behaves equivalently as a many-body system. An approximate formula for the time evolution of the reduced density matrix of a system interacting with a quantum chaotic environment is derived. This theoretical formulation is substantiated by the numerical study of the decoherence of two qubits interacting with a quantum chaotic environment modeled by a chaotic kicked top. Like the many-body model of environment, the quantum chaotic system is an efficient decoherer, and it can generate entanglement between the two qubits which have no direct interaction.
Floquet analysis of a fractal-spectrum-generating periodically driven quantum system
(APS, 2018-10) Bandyopadhyay, Jayendra N.; Sarkar, Tapomoy Guha
We employ Floquet analysis to study the spectral properties of a double-kicked top (DKT) system. This is a classically nonintegrable dynamical system, which also shows chaos. However, even for the underlying classically chaotic dynamics, the quantum quasienergy spectrum of this system does not follow the random matrix conjecture which was proposed for the quantum spectrum of any classically chaotic systems. Instead the quasienergy spectrum of the DKT system shows a butterfly-like self-similar fractal spectrum. Here we investigate the relation between the quasienergy spectrum and the energy spectrum of the corresponding time-independent Floquet Hamiltonian. This Hamiltonian is determined by factorizing the Floquet time-evolution operator into three terms: an initial kick and a final kick, and in between a time-independent evolution dictated by a time-independent Hermitian operator which is called the Floquet Hamiltonian. Like any other generic systems, the Floquet Hamiltonian of the DKT system is also not possible to determine exactly. We apply a recently proposed perturbation theory to obtain the approximate Floquet Hamiltonian at the high-frequency driving limit. We then study the parameter regime where the quasienergy spectrum of the Floquet time-evolution operator matches the energy spectrum of the approximate Floquet Hamiltonian. We have also done a comparative analysis of how the two butterfly spectra disappear with the variation of a system parameter. Finally, we also explore the self-similar property of the energy spectrum of the approximate Floquet Hamiltonian and find its connection with the Farey sequence. Unlike all previous studies, here we have extensively investigated the self-similar property of the whole DKT butterfly.
Floquet analysis of periodically driven kicked systems
(IAS, 2019) Bandyopadhyay, Jayendra N.
We employ Floquet theory to study the spectral properties of the Floquet Hamiltonian, also known as the effective static Hamiltonian of periodically driven kicked systems. In general, the Floquet Hamiltonian cannot be determined exactly, and therefore one has to employ some perturbation theory. Here we apply a recently proposed perturbation theory to obtain the Floquet Hamiltonian periodically kicked systems at very high-frequency limit. We studied the spectral properties of two well-known kicked systems: single and double-kicked top. Classical dynamics of these systems is chaotic, but their quantum mechanical spectrum is very different: the first one follows the Bohigas–Giannoni–Schmit conjecture of random matrix theory, but the latter one shows self-similar fractal behavior. Here we show that the fractal spectrum of the double-kicked top system shares some number of theoretical properties with the famous Hoftstadter butterfly.
Floquet analysis of pulsed Dirac systems: a way to simulate rippled graphene
(Springer, 2015-09) Sarkar, Tapomoy Guha; Bandyopadhyay, Jayendra N.
The low energy continuum limit of graphene is effectively known to be modeled using the Dirac equation in (2 + 1) dimensions. We consider the possibility of using a modulated high frequency periodic driving of a two-dimensional system (optical lattice) to simulate properties of rippled graphene. We suggest that the Dirac Hamiltonian in a curved background space can also be effectively simulated by a suitable driving scheme in an optical lattice. The time dependent system yields, in the approximate limit of high frequency pulsing, an effective time independent Hamiltonian that governs the time evolution, except for an initial and a final kick. We use a specific form of 4-phase pulsed forcing with suitably tuned choice of modulating operators to mimic the effects of curvature. The extent of curvature is found to be directly related to ω −1 the time period of the driving field at the leading order. We apply the method to engineer the effects of curved background space. We find that the imprint of curvilinear geometry modifies the electronic properties, such as LDOS, significantly. We suggest that this method shall be useful in studying the response of various properties of such systems to non-trivial geometry without requiring any actual physical deformations.
Floquet analysis of pulsed Dirac systems: a way to simulate rippled graphene
(Springer, 2015-09) Bandyopadhyay, Jayendra N.
The low energy continuum limit of graphene is effectively known to be modeled using the Dirac equation in (2 + 1) dimensions. We consider the possibility of using a modulated high frequency periodic driving of a two-dimensional system (optical lattice) to simulate properties of rippled graphene. We suggest that the Dirac Hamiltonian in a curved background space can also be effectively simulated by a suitable driving scheme in an optical lattice. The time dependent system yields, in the approximate limit of high frequency pulsing, an effective time independent Hamiltonian that governs the time evolution, except for an initial and a final kick. We use a specific form of 4-phase pulsed forcing with suitably tuned choice of modulating operators to mimic the effects of curvature. The extent of curvature is found to be directly related to ω −1 the time period of the driving field at the leading order. We apply the method to engineer the effects of curved background space. We find that the imprint of curvilinear geometry modifies the electronic properties, such as LDOS, significantly. We suggest that this method shall be useful in studying the response of various properties of such systems to non-trivial geometry without requiring any actual physical deformations.
Floquet engineering of Lie algebraic quantum systems
(APS, 2022-01) Bandyopadhyay, Jayendra N.
We propose a “Floquet engineering” formalism to systematically design a periodic driving protocol in order to stroboscopically realize the desired system starting from a given static Hamiltonian. The formalism is applicable to interacting and noninteracting quantum systems which have an underlying closed Lie algebraic structure. Unlike previous attempts at Floquet engineering, our method produces the desired Floquet Hamiltonian at any driving frequency and is not restricted to the fast or slow driving regimes. The approach is based on Wei-Norman ansatz, which was originally proposed to construct a time-evolution operator for any arbitrary driving. Here, we apply this ansatz to the micromotion dynamics, defined within one period of the driving, and engineer the functional form and operators of the driving protocol by fixing the gauge of the micromotion. To illustrate our idea, we use a two-band system or the systems consisting of two sublattices as a testbed. Particularly, we focus on engineering the cross-stitched lattice model that has been a paradigmatic flat-band model.
Floquet topological phase transitions in a kicked Haldane-Chern insulator
(APS, 2018-02) Bandyopadhyay, Jayendra N.; Sarkar, Tapomoy Guha
We consider a periodically δ-kicked Haldane type Chern insulator with the kicking applied in the ˆz direction. This is known to behave as an inversion symmetry breaking perturbation, since it introduces a time-dependent staggered sublattice potential. We study here the effects of such driving on the topological phase diagram of the original Haldane model of a Hall effect in the absence of a net magnetic field. The resultant Floquet band topology is again that of a Chern insulator with the driving parameters—frequency and amplitude— influencing the inversion breaking mass M of the undriven Haldane model. A family of such periodically related “Semenoff masses” is observed to occur, which support a periodic repetition of Haldane like phase diagrams along the inversion breaking axis of the phase plots. Out of these it is possible to identify two in-equivalent masses in the reduced zone scheme of the Floquet quasienergies, which form the centers of two inequivalent phase diagrams. Further, variation in the driving amplitude's magnitude alone is shown to effect the topological properties by linearly shifting the phase diagram of the driven model about the position of the undriven case, a phenomenon that allows the study of Floquet topological phase transitions in the system. Finally, we also discuss some issues regarding the modifications to Haldane's condition for preventing band overlaps at the Dirac point touchings in the Brillouin zone, in the presence of kicking.